Optimal. Leaf size=92 \[ \frac {(b d-a e)^3 (a+b x)^5}{5 b^4}+\frac {e (b d-a e)^2 (a+b x)^6}{2 b^4}+\frac {3 e^2 (b d-a e) (a+b x)^7}{7 b^4}+\frac {e^3 (a+b x)^8}{8 b^4} \]
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Rubi [A]
time = 0.09, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45}
\begin {gather*} \frac {3 e^2 (a+b x)^7 (b d-a e)}{7 b^4}+\frac {e (a+b x)^6 (b d-a e)^2}{2 b^4}+\frac {(a+b x)^5 (b d-a e)^3}{5 b^4}+\frac {e^3 (a+b x)^8}{8 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^3 \, dx\\ &=\int \left (\frac {(b d-a e)^3 (a+b x)^4}{b^3}+\frac {3 e (b d-a e)^2 (a+b x)^5}{b^3}+\frac {3 e^2 (b d-a e) (a+b x)^6}{b^3}+\frac {e^3 (a+b x)^7}{b^3}\right ) \, dx\\ &=\frac {(b d-a e)^3 (a+b x)^5}{5 b^4}+\frac {e (b d-a e)^2 (a+b x)^6}{2 b^4}+\frac {3 e^2 (b d-a e) (a+b x)^7}{7 b^4}+\frac {e^3 (a+b x)^8}{8 b^4}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(217\) vs. \(2(92)=184\).
time = 0.02, size = 217, normalized size = 2.36 \begin {gather*} a^4 d^3 x+\frac {1}{2} a^3 d^2 (4 b d+3 a e) x^2+a^2 d \left (2 b^2 d^2+4 a b d e+a^2 e^2\right ) x^3+\frac {1}{4} a \left (4 b^3 d^3+18 a b^2 d^2 e+12 a^2 b d e^2+a^3 e^3\right ) x^4+\frac {1}{5} b \left (b^3 d^3+12 a b^2 d^2 e+18 a^2 b d e^2+4 a^3 e^3\right ) x^5+\frac {1}{2} b^2 e \left (b^2 d^2+4 a b d e+2 a^2 e^2\right ) x^6+\frac {1}{7} b^3 e^2 (3 b d+4 a e) x^7+\frac {1}{8} b^4 e^3 x^8 \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(228\) vs.
\(2(84)=168\).
time = 0.66, size = 229, normalized size = 2.49
method | result | size |
norman | \(\frac {e^{3} b^{4} x^{8}}{8}+\left (\frac {4}{7} a \,b^{3} e^{3}+\frac {3}{7} b^{4} d \,e^{2}\right ) x^{7}+\left (a^{2} b^{2} e^{3}+2 d \,e^{2} a \,b^{3}+\frac {1}{2} d^{2} e \,b^{4}\right ) x^{6}+\left (\frac {4}{5} a^{3} b \,e^{3}+\frac {18}{5} a^{2} b^{2} d \,e^{2}+\frac {12}{5} d^{2} e a \,b^{3}+\frac {1}{5} b^{4} d^{3}\right ) x^{5}+\left (\frac {1}{4} e^{3} a^{4}+3 d \,e^{2} a^{3} b +\frac {9}{2} d^{2} e \,a^{2} b^{2}+a \,b^{3} d^{3}\right ) x^{4}+\left (d \,e^{2} a^{4}+4 d^{2} e \,a^{3} b +2 d^{3} a^{2} b^{2}\right ) x^{3}+\left (\frac {3}{2} d^{2} e \,a^{4}+2 d^{3} a^{3} b \right ) x^{2}+a^{4} d^{3} x\) | \(222\) |
default | \(\frac {e^{3} b^{4} x^{8}}{8}+\frac {\left (4 a \,b^{3} e^{3}+3 b^{4} d \,e^{2}\right ) x^{7}}{7}+\frac {\left (6 a^{2} b^{2} e^{3}+12 d \,e^{2} a \,b^{3}+3 d^{2} e \,b^{4}\right ) x^{6}}{6}+\frac {\left (4 a^{3} b \,e^{3}+18 a^{2} b^{2} d \,e^{2}+12 d^{2} e a \,b^{3}+b^{4} d^{3}\right ) x^{5}}{5}+\frac {\left (e^{3} a^{4}+12 d \,e^{2} a^{3} b +18 d^{2} e \,a^{2} b^{2}+4 a \,b^{3} d^{3}\right ) x^{4}}{4}+\frac {\left (3 d \,e^{2} a^{4}+12 d^{2} e \,a^{3} b +6 d^{3} a^{2} b^{2}\right ) x^{3}}{3}+\frac {\left (3 d^{2} e \,a^{4}+4 d^{3} a^{3} b \right ) x^{2}}{2}+a^{4} d^{3} x\) | \(229\) |
risch | \(\frac {1}{8} e^{3} b^{4} x^{8}+\frac {4}{7} x^{7} a \,b^{3} e^{3}+\frac {3}{7} x^{7} b^{4} d \,e^{2}+x^{6} a^{2} b^{2} e^{3}+2 x^{6} d \,e^{2} a \,b^{3}+\frac {1}{2} x^{6} d^{2} e \,b^{4}+\frac {4}{5} x^{5} a^{3} b \,e^{3}+\frac {18}{5} x^{5} a^{2} b^{2} d \,e^{2}+\frac {12}{5} x^{5} d^{2} e a \,b^{3}+\frac {1}{5} x^{5} b^{4} d^{3}+\frac {1}{4} a^{4} e^{3} x^{4}+3 x^{4} d \,e^{2} a^{3} b +\frac {9}{2} x^{4} d^{2} e \,a^{2} b^{2}+x^{4} a \,b^{3} d^{3}+a^{4} d \,e^{2} x^{3}+4 a^{3} b \,d^{2} e \,x^{3}+2 a^{2} b^{2} d^{3} x^{3}+\frac {3}{2} d^{2} e \,a^{4} x^{2}+2 x^{2} d^{3} a^{3} b +a^{4} d^{3} x\) | \(246\) |
gosper | \(\frac {x \left (35 e^{3} b^{4} x^{7}+160 x^{6} a \,b^{3} e^{3}+120 x^{6} b^{4} d \,e^{2}+280 x^{5} a^{2} b^{2} e^{3}+560 x^{5} d \,e^{2} a \,b^{3}+140 x^{5} d^{2} e \,b^{4}+224 x^{4} a^{3} b \,e^{3}+1008 x^{4} a^{2} b^{2} d \,e^{2}+672 x^{4} d^{2} e a \,b^{3}+56 x^{4} b^{4} d^{3}+70 x^{3} e^{3} a^{4}+840 x^{3} d \,e^{2} a^{3} b +1260 x^{3} d^{2} e \,a^{2} b^{2}+280 x^{3} a \,b^{3} d^{3}+280 a^{4} d \,e^{2} x^{2}+1120 a^{3} b \,d^{2} e \,x^{2}+560 a^{2} b^{2} d^{3} x^{2}+420 x \,d^{2} e \,a^{4}+560 x \,d^{3} a^{3} b +280 a^{4} d^{3}\right )}{280}\) | \(248\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 220 vs.
\(2 (86) = 172\).
time = 0.31, size = 220, normalized size = 2.39 \begin {gather*} \frac {1}{8} \, b^{4} x^{8} e^{3} + a^{4} d^{3} x + \frac {1}{7} \, {\left (3 \, b^{4} d e^{2} + 4 \, a b^{3} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b^{4} d^{2} e + 4 \, a b^{3} d e^{2} + 2 \, a^{2} b^{2} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{3} + 12 \, a b^{3} d^{2} e + 18 \, a^{2} b^{2} d e^{2} + 4 \, a^{3} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} d^{3} + 18 \, a^{2} b^{2} d^{2} e + 12 \, a^{3} b d e^{2} + a^{4} e^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} d^{3} + 4 \, a^{3} b d^{2} e + a^{4} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{3} + 3 \, a^{4} d^{2} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 227 vs.
\(2 (86) = 172\).
time = 2.87, size = 227, normalized size = 2.47 \begin {gather*} \frac {1}{5} \, b^{4} d^{3} x^{5} + a b^{3} d^{3} x^{4} + 2 \, a^{2} b^{2} d^{3} x^{3} + 2 \, a^{3} b d^{3} x^{2} + a^{4} d^{3} x + \frac {1}{280} \, {\left (35 \, b^{4} x^{8} + 160 \, a b^{3} x^{7} + 280 \, a^{2} b^{2} x^{6} + 224 \, a^{3} b x^{5} + 70 \, a^{4} x^{4}\right )} e^{3} + \frac {1}{35} \, {\left (15 \, b^{4} d x^{7} + 70 \, a b^{3} d x^{6} + 126 \, a^{2} b^{2} d x^{5} + 105 \, a^{3} b d x^{4} + 35 \, a^{4} d x^{3}\right )} e^{2} + \frac {1}{10} \, {\left (5 \, b^{4} d^{2} x^{6} + 24 \, a b^{3} d^{2} x^{5} + 45 \, a^{2} b^{2} d^{2} x^{4} + 40 \, a^{3} b d^{2} x^{3} + 15 \, a^{4} d^{2} x^{2}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 243 vs.
\(2 (80) = 160\).
time = 0.03, size = 243, normalized size = 2.64 \begin {gather*} a^{4} d^{3} x + \frac {b^{4} e^{3} x^{8}}{8} + x^{7} \cdot \left (\frac {4 a b^{3} e^{3}}{7} + \frac {3 b^{4} d e^{2}}{7}\right ) + x^{6} \left (a^{2} b^{2} e^{3} + 2 a b^{3} d e^{2} + \frac {b^{4} d^{2} e}{2}\right ) + x^{5} \cdot \left (\frac {4 a^{3} b e^{3}}{5} + \frac {18 a^{2} b^{2} d e^{2}}{5} + \frac {12 a b^{3} d^{2} e}{5} + \frac {b^{4} d^{3}}{5}\right ) + x^{4} \left (\frac {a^{4} e^{3}}{4} + 3 a^{3} b d e^{2} + \frac {9 a^{2} b^{2} d^{2} e}{2} + a b^{3} d^{3}\right ) + x^{3} \left (a^{4} d e^{2} + 4 a^{3} b d^{2} e + 2 a^{2} b^{2} d^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{4} d^{2} e}{2} + 2 a^{3} b d^{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 240 vs.
\(2 (86) = 172\).
time = 2.62, size = 240, normalized size = 2.61 \begin {gather*} \frac {1}{8} \, b^{4} x^{8} e^{3} + \frac {3}{7} \, b^{4} d x^{7} e^{2} + \frac {1}{2} \, b^{4} d^{2} x^{6} e + \frac {1}{5} \, b^{4} d^{3} x^{5} + \frac {4}{7} \, a b^{3} x^{7} e^{3} + 2 \, a b^{3} d x^{6} e^{2} + \frac {12}{5} \, a b^{3} d^{2} x^{5} e + a b^{3} d^{3} x^{4} + a^{2} b^{2} x^{6} e^{3} + \frac {18}{5} \, a^{2} b^{2} d x^{5} e^{2} + \frac {9}{2} \, a^{2} b^{2} d^{2} x^{4} e + 2 \, a^{2} b^{2} d^{3} x^{3} + \frac {4}{5} \, a^{3} b x^{5} e^{3} + 3 \, a^{3} b d x^{4} e^{2} + 4 \, a^{3} b d^{2} x^{3} e + 2 \, a^{3} b d^{3} x^{2} + \frac {1}{4} \, a^{4} x^{4} e^{3} + a^{4} d x^{3} e^{2} + \frac {3}{2} \, a^{4} d^{2} x^{2} e + a^{4} d^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.55, size = 208, normalized size = 2.26 \begin {gather*} x^4\,\left (\frac {a^4\,e^3}{4}+3\,a^3\,b\,d\,e^2+\frac {9\,a^2\,b^2\,d^2\,e}{2}+a\,b^3\,d^3\right )+x^5\,\left (\frac {4\,a^3\,b\,e^3}{5}+\frac {18\,a^2\,b^2\,d\,e^2}{5}+\frac {12\,a\,b^3\,d^2\,e}{5}+\frac {b^4\,d^3}{5}\right )+a^4\,d^3\,x+\frac {b^4\,e^3\,x^8}{8}+\frac {a^3\,d^2\,x^2\,\left (3\,a\,e+4\,b\,d\right )}{2}+\frac {b^3\,e^2\,x^7\,\left (4\,a\,e+3\,b\,d\right )}{7}+a^2\,d\,x^3\,\left (a^2\,e^2+4\,a\,b\,d\,e+2\,b^2\,d^2\right )+\frac {b^2\,e\,x^6\,\left (2\,a^2\,e^2+4\,a\,b\,d\,e+b^2\,d^2\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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